wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws
This addresses a limitation in PINNs for solving nonlinear hyperbolic PDEs with discontinuous solutions, which is incremental but important for computational physics applications.
The authors tackled the problem of approximating discontinuous entropy solutions of hyperbolic conservation laws, which standard Physics Informed Neural Networks (PINNs) fail to handle due to regularity requirements, by proposing weak PINNs (wPINNs) that accurately approximate these solutions, as demonstrated through numerical experiments.
Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by wPINNs and illustrate their performance through numerical experiments to demonstrate that wPINNs can approximate entropy solutions accurately.